6 May
2019
6 May
'19
11:43 a.m.
I probably should have asked if E(N) is monotonic *decreasing*, since E(1) = 1 and for N > 1, E(N) < 1. But after N = 2 it's hard to see what happens. Also, for N large almost all ellipses will not be near the boundary circle, so it seems likely that as N —> oo the configurations will approach close-packed disks in the plane, so the density will approach π/sqrt(12). —Dan ----- Let E(N) be the maximum possible fraction of a 2-disk D's area that is occupied by N non-overlapping, congruent ellipses in D. OK, the supremum. E(N) seems pretty hard to determine explicitly for each N. But one thing that seems not immediately obvious to me is: Question: --------- Is E(N) an increasing function of N ??? -----